FUNCTIONS AND MODELS
EXPONENTIAL FUNCTIONS
1.5 Exponential Functions
The function f(x) = 2x is called
an exponential function because
the variable, x, is the exponent.
ƒ It should not be confused with the power function
g(x) = x2, in which the variable is the base.
Exponential functions and their applications.
Exponential Functions
EXPONENTIAL FUNCTIONS
In general, an exponential function
is a function of the form f(x) = ax
where a is a positive constant.
If x = n, a positive integer, then:
If x = 0, then a0 = 1, and if x = -n,
where n is a positive
integer, then:
−n
a
=
1
an
If x is a rational number, x = p/q, where
p and q are integers and q > 0,
then:
ƒ Let’s recall what this means.
ax = a p/q = a p = ( a )p
q
q
EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
However, what is the meaning of ax
if x is an irrational number?
To help us answer that question,
we first look at the graph of the function
y = 2x, where x is rational.
ƒ For instance, what is meant by 2
3
or 5π ?
ƒ A representation of
this graph is shown here.
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EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
We want to enlarge the domain of y =
to include both rational and irrational
numbers.
2x
There are holes in the graph
corresponding to irrational values of x.
ƒ We want to fill in the holes
by defining f(x) = 2x,
where x ∈° , so that f is
an increasing function.
EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
In particular, since the irrational number
3 satisfies 1.7 < 3 < 1.8, we must
have 21.7 < 2 3 < 21.8
Similarly, if we use better approximations
for 3 , we obtain better approximations
for 2 3 :
1.73 < 3 < 1.74
⇒ 21.73 < 2
1.732 < 3 < 1.733
ƒ We know what 21.7 and 21.8
mean because 1.7 and 1.8
are rational numbers.
⇒ 21.732 < 2
1.73205 < 3 < 1.73206
21.7321,
21.73206,
…
< 21.7321
<2
3
< 21.73206
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
⋅
3
to be this number.
ƒ Using the preceding approximation process,
we can compute it correct to six decimal places:
2
and less than all the numbers
3
⋅
We define 2
21.7, 21.73, 21.732, 21.7320, 21.73205, …
⇒ 2
1.73205
< 21.733
⋅
⋅
It can be shown that there is exactly one
number that is greater than all the numbers
21.733,
3
⋅
EXPONENTIAL FUNCTIONS
21.74,
< 21.74
1.7320 < 3 < 1.7321 ⇒ 21.7320 < 2
EXPONENTIAL FUNCTIONS
21.8,
3
3
≈ 3.321997
ƒ Similarly, we can define 2x (or ax, if a > 0)
where x is any irrational number.
2
EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
The figure shows how all the holes in
the earlier figure have been filled to
complete the graph of the function
f(x) = 2x, x ∈° .
The graphs of members of the family of
functions y = ax are shown here for various
values of the base a.
EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
Notice that all these graphs pass through
the same point (0, 1) because a0 = 1 for a ≠ 0.
Notice also that as the base a gets larger,
the exponential function grows more rapidly
(for x > 0).
EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
You can see that there are basically three
kinds of exponential functions y = ax.
Those three cases are
illustrated here.
ƒ If 0 < a < 1, the exponential function decreases.
ƒ If a = 1, it is
a constant.
ƒ If a > 1,
it increases.
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EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
Observe that, if a ≠ 1, then the exponential
function y = ax has domain and range
(0, ∞ ).
Notice also that, since (1/a)x = 1/ax = a-x,
the graph of y = (1/a)x is just the reflection
of the graph of y = ax about the y-axis.
EXPONENTIAL FUNCTIONS
LAWS OF EXPONENTS
One reason for the importance of
the exponential function lies in the
following properties.
If a and b are positive numbers and
x and y are any real numbers, then:
ƒ If x and y are rational numbers, then these laws
are well known from elementary algebra.
ƒ It can be proved that they remain true for arbitrary
real numbers x and y.
EXPONENTIAL FUNCTIONS
Example 1
Sketch the graph of the function
y = 3 - 2x and determine its domain
and range.
1. ax + y = axay
2. ax – y = ax/ay
3. (ax)y = axy
4. (ab)x = axbx
EXPONENTIAL FUNCTIONS
Example 1
First, we reflect the graph of y = 2x
about the x-axis to get the graph of
y = -2x.
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EXPONENTIAL FUNCTIONS
Example 1
-2x
Then, we shift the graph of y =
upward 3 units to obtain the graph
of y = 3 - 2x (second figure).
ƒ The domain is
and the range
is (−∞, 3)
.
EXPONENTIAL FUNCTIONS
EXPONENTIAL FUNCTIONS
Example 2
Use a graphing device to compare
The exponential function f(x) = 2x and
the power function g(x) = x2.
Which function grows more quickly when x
is large?
Example 2
The figure shows both functions graphed
in the viewing rectangle [-2, 6] by [0, 40].
ƒ We see that the graphs intersect three times.
ƒ However, for x > 4, the graph of f(x) = 2x stays
above the graph of g(x) = x2.
EXPONENTIAL FUNCTIONS
Example 2
This figure gives a more global view
and shows that, for large values of x,
the exponential function y = 2x grows far
more rapidly than the power function y = x2.
APPLICATIONS OF EXPONENTIAL FUNCTIONS
APPLICATIONS OF EXPONENTIAL FUNCTIONS
The exponential function occurs
very frequently in mathematical models
of nature and society.
First, we consider a population of
bacteria in a homogeneous nutrient
medium.
ƒ Here, we indicate briefly how it arises in
the description of population growth.
ƒ In Chapter 3, we will pursue these and other
applications in greater detail.
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APPLICATIONS: BACTERIA POPULATION
APPLICATIONS: BACTERIA POPULATION
Suppose that, by sampling the population
at certain intervals, it is determined that
the population doubles every hour.
In general, p (t ) = 2t × 1000 = (1000)2t
ƒ If the number of bacteria at time t is p(t), where t
is measured in hours, and the initial population is
p(0) = 1000, then we have:
ƒ This population function is a constant multiple
of the exponential function.
ƒ So, it exhibits the rapid growth that we observed
in these figures.
p(1) = 2 p(0) = 2 ×1000
p(2) = 2 p (1) = 22 ×1000
p(3) = 2 p(2) = 23 ×1000
APPLICATIONS: BACTERIA POPULATION
APPLICATIONS
Under ideal conditions (unlimited space
and nutrition and freedom from disease),
this exponential growth is typical of what
actually occurs in nature.
What about the
human population?
APPLICATIONS: HUMAN POPULATION
APPLICATIONS: HUMAN POPULATION
The table shows data for the population of
the world in the 20th century.
The figure shows the corresponding scatter
plot.
The pattern of the data points
suggests exponential growth.
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APPLICATIONS: HUMAN POPULATION
APPLICATIONS: HUMAN POPULATION
So, we use a graphing calculator with
exponential regression capability to apply
the method of least squares and obtain the
exponential model
The figure shows the graph of this
exponential function together with
the original data points.
p = (0.008079266) ⋅ (1.013731)t
APPLICATIONS: HUMAN POPULATION
THE NUMBER e
We see that the exponential curve fits
the data reasonably well.
Of all possible bases for an exponential
function, there is one that is most convenient
for the purposes of calculus.
ƒ The period of relatively slow population growth
is explained by the two world wars and the Great
Depression of
the 1930s.
The choice of a base a is influenced by the
way the graph of y = ax crosses the y-axis.
THE NUMBER e
THE NUMBER e
The figures show the tangent lines to the
graphs of y = 2x and y = 3x at the point (0, 1).
If we measure the slopes of these
tangent lines at (0, 1), we find that
m ≈ 0.7 for y = 2x and m ≈ 1.1 for y = 3x.
ƒ Tangent lines will be defined precisely in Section 2.7.
ƒ For now, consider the tangent line to an exponential
graph at a point as the line touching the graph only at
that point.
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THE NUMBER e
THE NUMBER e
It turns out—as we will see in Chapter 3—
that some formulas of calculus will be greatly
simplified if we choose the base a so that
the slope of the tangent line to y = ax at (0, 1)
is exactly 1.
In fact, there is such a number and
it is denoted by the letter e.
THE NUMBER e
THE NUMBER e
In view of the earlier figures, it comes
as no surprise that:
In Chapter 3, we will see that
ƒ The number e lies between 2 and 3
ƒ The graph of y = ex lies between the graphs
of y = 2x and y = 3x
ƒ This notation was chosen by the Swiss
mathematician Leonhard Euler in 1727—probably
because it is the first letter of the word ‘exponential.’
the value of
e, correct to five decimal
places, is:
e ≈ 2.71828
THE NUMBER e
Graph the function
Example 3
1
y = e− x − 1
2
and state the domain and range.
THE NUMBER e
Example 3
We start with the graph of y = ex and
reflect about the y-axis to get the graph
of y = e-x.
ƒ Notice that the graph crosses the y-axis with a slope
of -1.
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THE NUMBER e
Example 3
THE NUMBER e
Example 3
Then, we compress the graph vertically
by a factor of 2 to obtain the graph
of y = 12 e − x (second figure).
Finally, we shift the graph downward
one unit to get the desired graph
(second figure).
THE NUMBER e
THE NUMBER e
How far to the right do you think we
would have to go for the height of the
graph of y = ex to exceed a million?
Use a graphing device to
find the values of x for which
ex > 1,000,000.
ƒ The domain is
and the range is (−1, ∞.)
Example 4
ƒ The next example demonstrates the rapid growth
of this function by providing an answer that might
surprise you.
THE NUMBER e
Example 4
Here, we graph both the function y = ex
and the horizontal line y = 1,000,000.
ƒ We see that these curves intersect when x ≈ 13.8
ƒ Thus, ex > 106 when x > 13.8
ƒ It is perhaps surprising
that the values of the
exponential function
have already surpassed
a million when x is only
14.
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